Pebble-Depth

TL;DR

This paper introduces pebble-depth, a new measure of sequence complexity based on pebble transducers, demonstrating its fundamental properties and comparing it with existing notions like finite-state, pushdown, and Lempel-Ziv depths.

Contribution

It defines pebble-depth using pebble transducers, establishes its core properties, and compares it with other depth measures, revealing unique sequence complexities.

Findings

Existence of a normal pebble-deep sequence without a normal finite-state-deep sequence.

A sequence with pebble-depth ~0.5 and Lempel-Ziv-depth ~0.

A sequence with pebble-depth ~1 and pushdown-depth ~0.5.

Abstract

In this paper we introduce a new formulation of Bennett's logical depth based on pebble transducers. This notion is defined based on the difference between the minimal length descriptional complexity of prefixes of infinite sequences from the perspective of finite-state transducers and pebble transducers. Our notion of pebble-depth satisfies the three fundamental properties of depth: i.e. easy sequences and random sequences are not deep, and the existence of a slow growth law type result. We also compare pebble-depth to other depth notions based on finite-state transducers, pushdown compressors and the Lempel-Ziv $78$ compression algorithm. We first demonstrate that there exists a normal pebble-deep sequence even though there is no normal finite-state-deep sequence. We then show that there exists a sequence which has pebble-depth level of roughly $1/2$ and Lempel-Ziv-depth level of roughly $0$. Finally we show the existence of a sequence which has a pebble-depth level of roughly $1$ and a pushdown-depth level of roughly $1/2$.